On some types of antinormal subgroups

A subgroup $H$ is called a pronormal subgroup in a group $G$ if for any $g \in G$ subgroups $H$ and $H^g$ are conjugated in ${<}H, H^g{>}$. A subgroup $H$ is called a contranormal subgroup in a group $G$ if $H^G = G$. A subgroup $H$ is called a weak pronormal subgroup in a group $G$ if $N_K(H)$ is contranormal in any subgroup $K$ which contains $H$. We obtained the examples of non pronormal but weak pronormal subgroups.

Structure of finite groups, in which any pronormal subgroup is either normal or abnormal

A subgroup $H$ of a group $G$ is said to be abnormal in $G$ if, for each element $g \in G$, we have $g \in {<}H, H^g{>}$. A subgroup $H$ of a group $G$ is said to be pronormal in $G$ if, for each element $g \in G$, the subgroups $H$ and $H^g$ are conjugate in ${<}H, H^g{>}$. We describe all finite groups, each pronormal subgroup in which is either normal or abnormal.

On transitive permutation groups with a subgroup satisfying a certain conjugacy condition

Let G be transitive permutation group of degree n and let K be a nontrivial pronormal subgroup of G (that is, for all g in G, K and Kg are conjugate in (K, Kg)). It is shown that K can fix at most ½(n – 1) points. Moreover if K fixes exactly ½(n – 1) points then G is either An or Sn, or GL(d, 2) in its natural representation where n = 2d-1 ≥ 7. Connections with a result of Michael O'Nan are dicussed, and an application to the Sylow subgroups of a one point stabilizer is given.

Prefrattini Subgroups and Cover-Avoidance Properties in 𝔘-Groups

W. Gaschutz [5] introduced a conjugacy class of subgroups of a finite soluble group called the prefrattini subgroups. These subgroups have the property that they avoid the complemented chief factors of G and cover the rest. Subsequently, these results were generalized by Hawkes [12], Makan [14; 15] and Chambers [2]. Hawkes [12] and Makan [14] obtained conjugacy classes of subgroups which avoid certain complemented chief factors associated with a saturated formation or a Fischer class. Makan [15] and Chambers [2] showed that if W, D and V are the prefrattini subgroup, 𝔍-normalizer and a strongly pronormal subgroup associated with a Sylow basis S, then any two of W, D and V permute and the products and intersections of these subgroups have an explicit cover-avoidance property.